Optical elements are pieces of substantially transparent material having surfaces that reflect or refract light, such as mirrors, lenses, splitters and collimators Optical elements are used in a variety of applications, including telescopes, microscopes, cameras and spectacles
Optical elements can be characterized by their optical properties and their surface optical properties. Optical properties such as astigmatism, optical power and prism describe how a wavefront of incident light is deformed as it passes through the optical element. Surface optical properties such as surface astigmatism, surface optical power and the gradients of the surface astigmatism and surface optical power describe geometrical properties of a surface of the optical element. The optical properties may be related to the surface optical properties.
Multifocal optical elements have more than one optical power. For example, bifocal lenses have two subregions, each with a different optical power. Bifocal spectacle lenses can be used, for example, to correct myopia (short-sightedness) in one subregion, and presbyopia (the loss of the eye's ability to change the shape of its lens) in the other subregion. Many people find it uncomfortable to wear bifocal spectacle lenses, because of the abrupt change in optical power from one subregion to the other. This led to the development of progressive spectacle lenses, which are multifocal lenses in which the optical power varies smoothly from one point to another on the lens.
The surface optical power at any point on the surface of an optical element is defined by the mean curvature of the surface. A progressive lens has varying optical power, so it has variable curvature, and is by definition aspherical. However, since the surface of the progressive lens, or at least a substantial part of it, is by definition aspherical, it has two distinct principal curvatures κ1 and κ2 at many points. The surface astigmatism at any point on the surface of an optical element is defined by the absolute value of the difference in the principal curvatures κ1 and κ2.
The definitions of the mean curvature H, Gaussian curvature G, and principal curvatures κ1 and κ2 of a surface ƒ at the point (x,y) are given in the following Equations 1A-1D:
                              H          =                                    1              2                        ⁢                                                                                                                                                        (                                                      1                            +                                                                                          (                                                                                                      ∂                                    f                                                                                                        ∂                                    x                                                                                                  )                                                            2                                                                                )                                                ⁢                                                  (                                                                                                                    ∂                                2                                                            ⁢                              f                                                                                      ∂                                                              y                                2                                                                                                              )                                                                    -                                              2                        ⁢                                                  (                                                                                    ∂                              f                                                                                      ∂                              x                                                                                )                                                ⁢                                                  (                                                                                    ∂                              f                                                                                      ∂                              y                                                                                )                                                ⁢                                                  (                                                                                                                    ∂                                2                                                            ⁢                              f                                                                                                                      ∂                                x                                                            ⁢                                                              ∂                                y                                                                                                              )                                                                    +                                                                                                                                                          (                                                  1                          +                                                                                    (                                                                                                ∂                                  f                                                                                                  ∂                                  y                                                                                            )                                                        2                                                                          )                                            ⁢                                              (                                                                                                            ∂                              2                                                        ⁢                            f                                                                                ∂                                                          x                              2                                                                                                      )                                                                                                                                          (                                      1                    +                                                                  (                                                                              ∂                            f                                                                                ∂                            x                                                                          )                                            2                                        +                                                                  (                                                                              ∂                            f                                                                                ∂                            y                                                                          )                                            2                                                        )                                                  3                  /                  2                                                                    ,                            (                  1          ⁢          A                )                                          G          =                                    1              2                        ⁢                                                                                (                                                                                            ∂                          2                                                ⁢                        f                                                                    ∂                                                  x                          2                                                                                      )                                    ⁢                                      (                                                                                            ∂                          2                                                ⁢                        f                                                                    ∂                                                  y                          2                                                                                      )                                                  -                                                      (                                                                                            ∂                          2                                                ⁢                        f                                                                                              ∂                          x                                                ⁢                                                  ∂                          y                                                                                      )                                    2                                                                              (                                      1                    +                                                                  (                                                                              ∂                            f                                                                                ∂                            x                                                                          )                                            2                                        +                                                                  (                                                                              ∂                            f                                                                                ∂                            y                                                                          )                                            2                                                        )                                2                                                    ,                            (                  1          ⁢          B                )                                                      κ            1                    =                      H            +                                                            H                  2                                -                G                                                    ,                            (                  1          ⁢          C                )                                          κ          2                =                  H          -                                                                      H                  2                                -                G                                      .                                              (                  1          ⁢          D                )            
Since the early days of designing progressive lenses, the main design goals have been to achieve:    a) gently varying optical power;    b) minimal astigmatism;    c) reduction of a variety of optical aberrations such as skew distortion, binocular imbalance, etc.
Many different methods have been proposed to achieve these goals. One such method is based on a base curve (meridian) runs from the upper part of the lens to its lower part. The lens surface is defined along the meridian such that the curvature varies gradually (and hence the optical power varies). Along the meridian itself, the principal curvatures κ1 and κ2 satisfy κ1=κ2. The lens surface is extended from the meridian horizontally in several different methods. Explicit formulas are given for the extensions from the meridian. Maitenaz obtains an area in the upper part of the lens, and another area in the lower part of the lens in which there is a rather stable optical power. Furthermore, the astigmatism in the vicinity of the meridian is relatively small.
Some designs for progressive lenses explicitly divide the progressive lens into three zones: an upper zone for far vision, a lower zone for near vision, and an intermediate zone that bridges the first two zones. The upper and lower zones provide essentially clear vision. Many designs use spherical surfaces for the upper and lower zones. A major effort in the design process is to determine a good intermediate zone.
Some design methods use explicit formulae to define the intermediate zone, to achieve a relatively smooth transition area between the upper and lower zones. In one existing method, the lens designer defines the value of the lens surface in the intermediate zone at a number of special points, which may relate to lens-related characteristics, e.g., a size of a frame intended to accommodate the lens, and/or a refraction index of the material of the lens. The full surface is then generated by the method of splines. The designer adjusts the value of the lens surface at the special points in order to improve the properties of the generated surface.
In yet another existing method, a base surface function is used to define a progressive lens having an upper zone for far vision, a lower zone for near vision, and an intermediate zone. According to this method, an improved progressive lens is calculated by optimizing a function defined over a subregion of the lens, where the optimized function is to be added to the base surface function.
The lens may be produced using different methods, for example, a freeform manufacturing process. The free-form manufacturing process may include using a Computer Numerically Controlled (CNC) milling apparatus, which may be capable of producing a lens having a desired surface geometry, at a relatively high degree of accuracy, based on input data defining the desired lens surface geometry.
The Free-form manufacturing process may include a generating stage and a polishing stage. The generating stage may include lathing/grinding a semi finished blank to produce an intermediate surface. Since the free-form process may be used for producing any desired optical surface at a relatively high degree of accuracy, the freeform manufacturing process may be applied to a spherical blank, which may be less expensive than progressive semi-finished blanks, e.g., such as those used in other lens manufacturing processes. In addition, a relatively small number of different blanks may be used for producing a relatively wide range of optical surfaces. The polishing stage may include polishing the intermediate surface in accordance with the required surface geometry and/or cosmetic requirements. The free-form process may enable using a relatively small number of polishing tools to produce a relatively wide range of optical surfaces.